This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem isline4N

Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses isline4.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
isline4.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
isline4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
isline4.n ⊢ 𝑁 = ( Lines ‘ 𝐾 )
isline4.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
Assertion isline4N ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) ∈ 𝑁 ↔ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) )

Proof

Step Hyp Ref Expression
1 isline4.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 isline4.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3 isline4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 isline4.n ⊢ 𝑁 = ( Lines ‘ 𝐾 )
5 isline4.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
6 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7 1 6 3 4 5 isline3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) ∈ 𝑁 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
8 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ HL )
9 1 3 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
10 9 adantl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ∈ 𝐵 )
11 simplr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
12 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
13 1 12 6 2 3 cvrval3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑝 𝐶 𝑋 ↔ ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ) )
14 8 10 11 13 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 𝐶 𝑋 ↔ ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ) )
15 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
16 15 ad3antrrr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → 𝐾 ∈ AtLat )
17 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → 𝑞 ∈ 𝐴 )
18 simplr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → 𝑝 ∈ 𝐴 )
19 12 3 atncmp ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑞 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴 ) → ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ↔ 𝑞 ≠ 𝑝 ) )
20 16 17 18 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ↔ 𝑞 ≠ 𝑝 ) )
21 necom ⊢ ( 𝑞 ≠ 𝑝 ↔ 𝑝 ≠ 𝑞 )
22 20 21 bitrdi ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ↔ 𝑝 ≠ 𝑞 ) )
23 eqcom ⊢ ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ↔ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) )
24 23 a1i ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ↔ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) )
25 22 24 anbi12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ 𝐴 ) → ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ↔ ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
26 25 rexbidva ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑝 ∧ ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) = 𝑋 ) ↔ ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
27 14 26 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 𝐶 𝑋 ↔ ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
28 27 rexbidva ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ) ) )
29 7 28 bitr4d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) ∈ 𝑁 ↔ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) )